List of practice Questions

Let $A = \{1, 2, 3\}$ and $B = \{a, b\}$ then number of relations from set $A$ to set $B$ are

Let $A$ and $B$ be two sets having $m$ and $n$ elements respectively, then total number of functions from $A$ to $B$ are

Match List-I with List-II. Suppose $A, B, C$ are three non-empty sets.

List-I:

A. De-Morgan's Law,
B. Associative Law,
C. Distributive Law,
D. Difference of sets Law.

Match List-I with List-II. Suppose $A,B,C$ are three non-empty sets. List-I:
A. De-Morgan's Law,
B. Associative Law,
C. Distributive Law,
D. Difference of sets Law.

Given below are two statements:

Assertion (A):
The angle between the pair of lines \[ \frac{x+3}{3}=\frac{y-1}{5}=\frac{z+3}{4} \quad \text{and} \quad \frac{x+1}{1}=\frac{y-4}{1}=\frac{z-5}{2} \] is $\cos^{-1}\left(\frac{8\sqrt{3}}{15}\right)$.
Reason (R):
The angle between the two lines is \[ \cos\theta=\frac{a_1a_2+b_1b_2+c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}. \]

Let $A = \{1, 4, 7\}$ and $B = \{2, 3, 8\}$. Then the number of elements in the relation $R = \{ ((a_1, b_1), (a_2, b_2)) \in (A \times B) \times (A \times B) : a_1+b_2 \text{ divides } a_2+b_1 \}$ is :}

Let $ R=\{(x,y)\in \mathbb{N}\times\mathbb{N}:\log_e(x+y)\le2\} $. Then the minimum number of elements required to be added in $R$ to make it a transitive relation is ________.

Let \[ S=\{x\in[-\pi,\pi]:\sin x(\sin x+\cos x)=a,\; a\in\mathbb{Z}\}. \] Then $n(S)$ is equal to:

The relation $R$ on the set of real numbers defined by \[ R=\{(a,b):a\leq b^2\} \] is:
• [(A)] Reflexive
• [(B)] Not symmetric
• [(C)] Neither reflexive nor transitive
• [(D)] Transitive Choose the correct answer from the options given below:

If \[ R=\{(x,y)\mid x,y\in \mathbb{R},\ x^2+y^2=1\} \] is a relation in $\mathbb{R}$, then $R$ is:

Consider the relation R on the set $\{-2, -1, 0, 1, 2\}$ defined by $(a, b) \in R$ if and only if $1 + ab>0$. Then, among the statements:
I. The number of elements in R is 17
II. R is an equivalence relation

Let $A = \{1, 2, 3, 4, 5\}$ and $B = \{a, b, c\}$ then total number of functions from $A$ to $B$ which are not onto are

Let $ A = \{2,3,4,5,6\} $ be a set. Consider $ R $ be a relation on $ A \times A $ such that $ (x,y) \, R \, (a,b) $ implies $ x $ divides $ a $ and $ y \leq b $, then total number of elements in $ R $ is:

If $ A = \{1,2,3,4,5,6\} $ and $ B = \{1,2,3,\ldots,9\} $, then the number of strictly increasing functions $ f : A \to B $ such that $ f(i) \neq i $ for all $ i = 1,2,3,4,5,6 $ is:

The speed-density relation on a one-way, single lane road is shown in the figure, where speed $ u $ is in km/hour and density $ k $ is in vehicles/km. The maximum flow (in vehicles/hour) on this road is

Relation $ R = \{(x, y): x<y^2 \text{ where } x, y \in \mathbb{R}\} $ is:

Consider the following relation $ R = \{(4,5), (5,4), (7,6), (6,7)\} $ on set $ I = \{4,5,6,7\} $. Which of the following properties relation $ R $ does not have? (A) Reflexive property
(B) Symmetric property
(C) Transitive property
(D) Antisymmetric property
Choose the correct answer from the options given below:

In a group of 50 people, 30 like tea, 25 like coffee, and 10 like neither. How many like both tea and coffee?

Let $ A = \{-3, -2, -1, 0, 1, 2, 3\} $. A relation $ R $ is defined such that $ xRy $ if $ y = \max(x, 1) $. The number of elements required to make it reflexive is $ l $, the number of elements required to make it symmetric is $ m $, and the number of elements in the relation $ R $ is $ n $. Then the value of $ l + m + n $ is equal to:

How many combinations of non-null sets A, B, C are possible from the subsets of {2, 3, 5} satisfying the conditions: (i) A is a subset of B, and (ii) B is a subset of C?

If the set $R = {(a, b) : a + 5b = 42, a, b \in \mathbb{N}}$ has $m$ elements and $\sum_{n=1}^m (1 + i^n) = x + iy$, where $i = \sqrt{-1}$, then the value of $m + x + y$ is:

An organization awarded 48 medals in event 'A', 25 in event 'B' and 18 in event 'C'. If these medals went to total 60 men and only five men got medals in all the three events, then how many received medals in exactly two of three events?

Let $A = \{x \in \mathbb{R}: |x+3| + |x+4| \le 3\}$,} \[ B = \left\{ x \in \mathbb{R} : 3 \cdot \sum_{r=1}^{\infty} \frac{3^{x-3}}{10^r} < 3^{-3x} \right\}, \] where $[x]$ denotes the greatest integer function. Then,